Clifford's theorem on special divisors
In mathematics, Clifford's theorem on special divisors is a result of William K. Clifford (1878) on algebraic curves, showing the constraints on special linear systems on a curve C.
Statement
A divisor on a Riemann surface C is a formal sum [math]\displaystyle{ \textstyle D = \sum_P m_P P }[/math] of points P on C with integer coefficients. One considers a divisor as a set of constraints on meromorphic functions in the function field of C, defining [math]\displaystyle{ L(D) }[/math] as the vector space of functions having poles only at points of D with positive coefficient, at most as bad as the coefficient indicates, and having zeros at points of D with negative coefficient, with at least that multiplicity. The dimension of [math]\displaystyle{ L(D) }[/math] is finite, and denoted [math]\displaystyle{ \ell(D) }[/math]. The linear system of divisors attached to D is the corresponding projective space of dimension [math]\displaystyle{ \ell(D)-1 }[/math].
The other significant invariant of D is its degree d, which is the sum of all its coefficients.
A divisor is called special if ℓ(K − D) > 0, where K is the canonical divisor.[1]
Clifford's theorem states that for an effective special divisor D, one has:
- [math]\displaystyle{ 2(\ell(D)- 1) \le d }[/math],
and that equality holds only if D is zero or a canonical divisor, or if C is a hyperelliptic curve and D linearly equivalent to an integral multiple of a hyperelliptic divisor.
The Clifford index of C is then defined as the minimum of [math]\displaystyle{ d - 2(\ell(D) - 1) }[/math] taken over all special divisors (except canonical and trivial), and Clifford's theorem states this is non-negative. It can be shown that the Clifford index for a generic curve of genus g is equal to the floor function [math]\displaystyle{ \lfloor\tfrac{g-1}{2}\rfloor. }[/math]
The Clifford index measures how far the curve is from being hyperelliptic. It may be thought of as a refinement of the gonality: in many cases the Clifford index is equal to the gonality minus 2.[2]
Green's conjecture
A conjecture of Mark Green states that the Clifford index for a curve over the complex numbers that is not hyperelliptic should be determined by the extent to which C as canonical curve has linear syzygies. In detail, one defines the invariant a(C) in terms of the minimal free resolution of the homogeneous coordinate ring of C in its canonical embedding, as the largest index i for which the graded Betti number βi, i + 2 is zero. Green and Robert Lazarsfeld showed that a(C) + 1 is a lower bound for the Clifford index, and Green's conjecture states that equality always holds. There are numerous partial results.[3]
Claire Voisin was awarded the Ruth Lyttle Satter Prize in Mathematics for her solution of the generic case of Green's conjecture in two papers.[4][5] The case of Green's conjecture for generic curves had attracted a huge amount of effort by algebraic geometers over twenty years before finally being laid to rest by Voisin.[6] The conjecture for arbitrary curves remains open.
Notes
- ↑ Hartshorne p.296
- ↑ Eisenbud (2005) p.178
- ↑ Eisenbud (2005) pp. 183-4.
- ↑ Green's canonical syzygy conjecture for generic curves of odd genus - Claire Voisin
- ↑ Green’s generic syzygy conjecture for curves of even genus lying on a K3 surface - Claire Voisin
- ↑ Satter Prize
References
- Arbarello, Enrico; Cornalba, Maurizio; Griffiths, Phillip A.; Harris, Joe (1985). Geometry of Algebraic Curves Volume I. Grundlehren de mathematischen Wisenschaften 267. ISBN 0-387-90997-4.
- Clifford, William K. (1878), "On the Classification of Loci", Philosophical Transactions of the Royal Society of London (The Royal Society) 169: 663–681, doi:10.1098/rstl.1878.0020, ISSN 0080-4614
- Eisenbud, David (2005). The Geometry of Syzygies. A second course in commutative algebra and algebraic geometry. Graduate Texts in Mathematics. 229. New York, NY: Springer-Verlag. ISBN 0-387-22215-4.
- Fulton, William (1974). Algebraic Curves. Mathematics Lecture Note Series. W.A. Benjamin. p. 212. ISBN 0-8053-3080-1.
- Griffiths, Phillip A.; Harris, Joe (1994). Principles of Algebraic Geometry. Wiley Classics Library. Wiley Interscience. p. 251. ISBN 0-471-05059-8.
- Hartshorne, Robin (1977). Algebraic Geometry. Graduate Texts in Mathematics. 52. ISBN 0-387-90244-9.
External links
- Hazewinkel, Michiel, ed. (2001), "Clifford theorem", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4, https://www.encyclopediaofmath.org/index.php?title=C/c022490
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